This talk focus on the inverse obstacle problem for the Laplace operator, with an obstacle characterized by a Dirichlet condition, that is surely the most simple geometric inverse problem possible. Using this toy problem, I will present the typical questions we are interested in in the Inverse Problems for Pdes community, and the techniques I have developed with lots of collaborators to tackle them. Some numerical reconstructions will be presented to show the efficiency of the proposed reconstruction methods.
I will talk about clause learning and its application to solving the graph coloring problem. The graph coloring problem asks for the smallest number of colors we can assign to the vertices of a graph so that no two adjacent vertices take the same color. It is a major component of numerous allocation and scheduling problems.
I will present the approach that we recently proposed with George Katsirelos. This method uses the symmetry-free Zykov tree to explore the solution space, prunes this tree with a novel dual bound from finding embedded Mycielskian graphs, and computes conflicts during search in order to learn from past failures.
The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, a Gaussian convergence can be established for the procedure. Here, we are interested in the local limit theorem, that is, quantifying this convergence on the density of the involved objects. The analysis relies on a parametrix technique for Markov chains converging to diffusions, where the drift is unbounded.
Partant du filtre de Kalman conventionnel pour l'estimation d'état de systèmes linéaires à temps discret avec hypothèses gaussiennes, le filtre de Kalman intervalle (IKF, 1997) et son amélioration (iIKF, 2013), sont présentés. Ces derniers permettent d’intégrer des incertitudes à erreurs bornées aux incertitudes statistiques déjà gérées par le filtre conventionnel. On peut en particulier représenter des bruits gaussiens incertains, c’est-à-dire de moyenne et matrice de covariance définies par leur appartenance à des intervalles ou matrices intervalles.
Les principaux inconvénients de ces deux approches à base d'intervalles sont présentés et un nouveau filtre, noté UBIKF (Minimum Upper Bound of Variance Interval Kalman Filter, 2017) est décrit. Il repose sur la recherche d’une matrice de gain réelle (i.e. non incertaine) minimisant une borne majorante de l’ensemble des matrices de covariance de l’erreur d’estimation en respectant les bornes des incertitudes paramétriques. Un encadrement de tous les estimés possibles est ensuite déterminé en utilisant l’analyse par intervalles. Le filtre UBIKF permet de réduire à la fois la complexité calculatoire de l’inversion ensembliste des matrices intervalles présentes dans le filtre iIKF et le conservatisme des estimations.
The Koopman operator framework is an increasingly popular tool for data-driven analysis of complex nonlinear dynamical systems that emerged in the early 2000s. This talk will briefly review this framework, describe a recent extension to control and discuss connections to machine learning. The main idea of the Koopman operator framework is to equivalently represent a nonlinear system by an infinite-dimensional linear operator, thereby trading off nonlinearity for infinite-dimensionality. After suitable truncation, this allows for spectral methods and other linear techniques to be deployed for analysis of the nonlinear system in a spirit similar to the classical linear systems theory.
When extended to systems with control inputs, a truncation of this operator results in a linear predictor whose state evolves on a higher-dimensional (lifted or embedded) state-space. This allows one to design controllers for the nonlinear dynamical system using linear control design methods; in particular, we show that standard linear model predictive control can be designed with computational complexity of the underlying optimization problem independent of the dimension of the embedded state-space, thereby allowing for deployment of the controller even under tight computation time constraints. The operator truncation procedure works directly with observed trajectories of the dynamical system and hence the entire scheme is data-driven. The approach will be demonstrated on numerical examples including power grid and PDE control, and its theoretical properties will be discussed.